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A Josephson junction is a structure consisting of two superconducting electrodes separated by a thin insulating layer, usually of the order of 2 nanometers in thickness. This structure is named after B.D. Josephson who predicted¹ the existence of a small current without any voltage drop. The junction is partly characterized by its Josephson coupling energy , where I_C is the critical current, which is the maximum zero-voltage current of the junction. If the junction is sufficiently small the charging energy , where C is the capacitance of the junction, becomes important. This is the energy stored in the electric field of a capacitor C with the charge ±e on its electrodes. Thus if a Cooper pair tunnels in an uncharged Josephson junction it will momentarily charge the junction with the charge ±2e, increasing the energy by 4E_C. In a single junction this is usually not a problem because with some help from Heisenberg's uncertainty law, Et>/2, the charge slips quickly away from the junction, discharging it before the energy conservation law notices. But if there is any obstacle close after the junction, for example another junction or a resistor, the junction might stay charged long enough for the energy law to catch up. Then the tunneling will be suppressed and there will be no current without a little voltage to help. This is called the Coulomb blockade² of Cooper pair tunneling. In a simple model, t is the discharge time for the capacitor, t=R_eC, where R_e is the impedance of the junction environment, assumed frequency independent. The condition for Coulomb blockade is then 4E_C>E>/2t, which implies R_e>/4e²1 k.

In an array of Josephson junctions the tunneling behavior is partly determined by the ratio of E_J to E_C. If E_J >>E_C there will be no Coulomb blockade and the array will be superconducting. If, on the other hand, E_C>>E_J, there will be a clear Coulomb blockade.

If a magnetic field is applied perpendicular to the array the behavior can usually be described in terms of frustration^3,4, provided that the array is homogeneous, i.e. the junctions, holes and superconducting "islands" are the same in the entire array, and provided that the array is considerably smaller than the Josephson penetration depth⁵ (several mm in this case). Frustration is dimensionless and proportional to magnetic field. Frustration f=1 corresponds to one flux quantum, ₀=h/2e, in every loop in the array. Properties that depend on frustration are periodic (f -> f+1) for sufficiently low frustrations. At high enough fields other effects come into account, like the suppression of superconductivity.

One interesting effect is the frustration dependence of the zero-bias resistance (the resistance close to zero current), which decreases at integer as well as rational frustrations (1/2, 1/3, 2/3 etc.). The resistance can change by five orders of magnitude, just by changing the magnetic field slightly^6,7. The threshold voltage also changes with frustration, as one measurement in this report shows. The threshold voltage is the minimum voltage required to start a current in an array with Coulomb blockade.

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